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Allow the total area and particles to divide between phases: $A=A_1+A_2$, $N=N_1+N_2$. Minimize $F_1(N_1,A_1,T)+F_2(N_2,A_2,T)$ subject to these constraints. Transferring area requires equal pressure and transferring particles requires equal chemical potential:Equivalently, take the lower convex envelope of free energy, using a common-tangent construction for phase coexistence. The lever rule fixes phase fractions between coexistence densities. Since the hexagonal branch lies below the square branch throughout their common domain here, this particular comparison supplies no stable square–hexagonal coexistence. A separate fluid free energy is needed to predict fluid–ordered coexistence and its endpoints.

For the actual two-dimensional hard-disc system, a hexatic phase can intervene between isotropic fluid and crystal; the one-particle cell model cannot describe that orientational-order physics. It is therefore safe to predict competition between fluid and ordered phases, but not a precise two-phase interval or a direct fluid–crystal transition from these two crystal branches alone.